Updated: January 23, 2008
Raman Scattering from Thermal Fluctuations
of an Elastic Sphere in an Elastic Matrix

........

  

The Clausius-Mossotti relation:
(deps/drho)= (eps-1)(eps+2)/(3rho) --- rho = mass density, eps = permittivity ph.utexas.edu
(εr-1)/(εr+2) = (n α)/(3 ε0)
where n is number of atoms per cubic metre.   [I suspect this is correct in SI units]
ref
AC Electrokinetics: Applications for Nanotechnology ref
(εr-1)/(εr+2) = NαE/(3 ε0) xxx
Lorenz-Lorenz equation
n2-1
---------
n2+2		 = N α
--------
3 ε0V
http://chsfpc5.chem.ncsu.edu/CH795Z/lecture/lecture10/dielectric_polarization.html
References

J. Jeong, S. Shin, I. Lyubchanskii and V. Varyukhin "Strain-induced three-photon effects" Phys. Rev. B vol. 62, no. 20 (2000) They call pijkl the "linear photoelastic tensor"
cnsm.kaist.ac.kr ::
They write εij = ε0ij + pijklukl, but this definition of p is not common. They don't say which definition of the strain tensor they use.

S. Sinha, K. Urbanek "Properties and implementation of acousto-optic superlattice tunable filters in fibers" They call pijkl the "strain optic tensor". stanford.edu/.../11-22-02.doc

Mention of strain optic tensor p:
http://smartech.kaist.ac.kr/PROJECT/smart/FOS.htm
For "optical fiber" they use the values refractive index nc = 1.456, ν=0.17, p11 = 0.113, p12 = 0.252.

L. Hansen "Constant Frequency Condition of Fiber Lasers in Strain" 15th Nordic Seminar on Computational Mechanics (2002) ime.auc.dk ::
They mention "the photoelastic effect" and also mention the "strain optic tensor" pijkl, but it is for "impermeability" (inverse of permittivity). For isotropic materials we have p11 and p12.
p11 = 0.121 and p12 = 0.27 for standard optical fibers. They refer to the "impermeability tensor" ηij, which is 1/n2.
(1/n2)ij = (1/n2) + pijkl εkl where εkl is the 3×3 strain tensor. But they don't say which definition of the strain tensor is used.


p11 and p12 are called Pockel's coefficients of the stress-optic tensor "Coupled together, the refractive index, Pockel�s coefficients, and Poisson�s ratio are known as the photoelastic coefficient" pe sensorsmag.com

P. Williams et al. "Optical, thermo-optic... Bi4Ge3O12" Applied Optics, vol. 35 (1996) pages 3562-3569. (a cubic crystal of symmetry 43m) They don't say which definition of the strain tensor they use. boulder.nist.gov/.../Williams-AO-96.pdf ::

J. Mansell et al. "Evaluating the effect of transmissive optic thermal lensing on..." Applied Optics vol. 40, no. 3 (2001) pages 366-374 stanford.edu :: They mention a "photoelastic coefficient" called "ρ12" (ρ rather than p), and give values. But they don't say which definition of strain they use.

Table I:    Properties of some optical materials
Materialoptical
   n		p11p12pePoisson
ratio
  ν		 
Schott NG5 glass 1.5??? 0.27?????? [used in optical filters (source ::)]
fused silica1.5 ??? 0.270??????(source ::) 
Hoya FR5 Faraday Glass 1.7??? 0.26?????? [made by Hoya Company (source ::)]
TGG1.95??? 0.1?????? [Terbium Gallium Garnet (source ::)]
silica???0.120.27?????? (scholar.lib.vt.edu)
standard optical fibers???0.1210.27??????(ime.auc.dk ::)
dense amorphous SiO21.45????????????(userweb.mrl.uiuc.edu)
silica?????????0.22???(av.it.pt)
optical fiber1.4560.1130.252???0.17(smartech.kaist.ac.kr)

    A good general introduction to the discovery by Brewster, early history and engineering applications of photoelasticity is "Recent Advances in Photoelastic Applications" (ntu.edu.sg).

The photoelastic constants for silica are p11 = 0.12 and p12 = 0.27 scholar.lib.vt.edu They seem to be making a mistake in their equations (3.1a) (3.1b) (3.1c) since they write
nx - n = (-n3/2)( p11 σx + p12 σy + p12 σz )
but this is wrong on dimensional grounds. They say σx, σy... are diagonal elements of the stress tensor. nx is the refractive index along the x direction. Perhaps they say "stress" even though they mean "strain". In any case, they don't say which definition of strain they use.


Catalog of Optical Materials http://www.optotl.ru/MatEng.htm Basic physico-chemical characteristics of materials http://www.optotl.ru/SiEng.htm
MgO (NaCl structure):
p11 = -0.21 or -0.31 or -0.259
p12 = 0.04 or -0.07 or -0.011
p44 = -0.10 or -0.105

Tellurium Dioxide
p11 = 0.0074
p12 = 0.187
p13 = 0.340
p31 = 0.0905 (?)
p33 = 0.240
p66 = -0.0463
www.crystaltechnology.com/Tellurium_Dioxide.pdf

http://www.eps.org/aps/meet/MAR00/baps/abs/S4180001.html Photoelasticity of \alpha-quartz from first principles F. Detraux, X. Gonze Within Density Functional Perturbation Theory, we have studied the photoelasticity of SiO_2 quartz. This fourth rank tensor property describes the variation of the refractive index caused by strain. In the case of quartz, the photoelastic tensor contains 8 independent coefficients.
"Low photoelastic constant glass" http://www.ohara-inc.co.jp/b/b02/b0205_pb/b0205.htm
Photoelastic constant β (nm/cm/105 Pa) 0.02

http://www.ohara-gmbh.de/e/news/news.html
PBH55 glass -Low Photoelastic Constant (β) Glass

http://www.ohara-gmbh.com/e/katalog/tinfo_5_4.html
Photoelastic constant β:
δ = β × d × F
F = stress (Pa)
δ = optical path difference (nm)
d = thickness of glass (cm)

www.bellexinternational.com/Cytop%20Flyer.pdf
Photoelastic Constant 6.4 (Brewster)

http://www.sciner.com/Crystals/bism.htm
p11 0.12 or 0.13
p12 0.04 or 0.04

Anisotropy of the Elastooptic Properties of SLA and SLG Crystals ... crystalresearch.com ::

www.lasermaterials.com/LMC_Graphics/ LMC_Brochure_Summer_99.pdf
Nd:YAG
p11 = -0.029
p12 = 0.0091
p44 = -0.0615

"Faraday isolators for high average power: achieved results and..." [not much to see here] xsdf ::

M. Brito et al. "Stress in silicon ribbons crystallised from a molten zone: A study of the influence of growth parameters"
n1-n2= C(σ1-σ2)
n are principal components of index of refraction. σ are principal components of the stress tensor. C = photoelastic coefficient (stress-optic constant). C = 20 Br, valid for silicon crystal in [100] orientation. Polycrystalline silicon ribbons.
correio.cc.fc.ul.pt ::

Problem set #12 for Illinois MatSE 204: userweb.mrl.uiuc.edu

rpl.stanford.edu (link does not work) pe = (n2/2)(p12-ν(p11+p12))
pe = photoelastic coefficient
p11, p12 are Pockel's coefficients of the strain-optic tensor
ν is the Poisson ratio
pe is approximately 0.22

courses.washington.edu Stress optical coefficient is sometimes called f and measured in units of (MPa-mm/fringe).

G.-M. Rignanese, J.-P. Michenaud, X. Gonze "Ab initio study of the volume dependence of dynamical and thermodynamical properties of silicon" pdf [couldn't download pdf]

Tieyu Zheng and Steven Danyluk
"Study of stresses in thin silicon wafers with near-infrared phase stepping photoelasticity" mrs.org [pdf is for sale]

S.P. Wong, W.Y. Cheung, N. Ke, M.R. Sajan, W.S. Guo, L. Huang, Shounan Zhao,
"Infrared Photoelasticity Study of Stress Distribution in Silicon under
Thin Film Structures", Materials Chemistry and Physics (in press)
http://www.ee.cuhk.edu.hk/ee/staff/spwong.html
Email: [email protected]

It is the induced dipole moment p that generates the scattered radiation.
   The target object under study could be an atom, a molecule, an isolated nanoparticle or even a bulk sample. If α does not vary with time, then the scattered radiation and the incident radiation will have exactly the same frequency. This is called "Rayleigh scattering." If α varies sinusoidally in time at some frequency ω1, the scattered radiation can have two possible frequencies:
ωinc + ω1   (called "Stokes")
ωinc - ω1   (called "anti-Stokes")
   In either case, this is called "Raman scattering." An experiment to observe Raman scattering simply requires a high quality filter that rejects all scattered radiation with the same frequency as the incident beam.
   The polarizability tensor α can vary for different reasons. The simplest situation is a diatomic molecule (like N2 or O2) where α rotates as the molecule rotates. If the molecule rotates 200 times a second, a given component of α will rise and fall 400 times a second. Thus, the Raman shift is twice the frequency of rotation. Since quantum mechanics quantizes the allowed frequencies with which an object can rotate, a discrete Raman spectrum can be observed.
   If an object (atom, molecule or nanoparticle) undergoes mechanical vibrations, α will oscillate at the same frequency.
   Another important situation is where an object (atom, etc.) undergoes an electronic transition which causes α to change. The frequency of the oscillation of α is simply related to the energy difference between the two electronic levels.
   In some situations, an object can vibrate at some frequency without having a corresponding peak in the Raman spectrum. Some modes are not "Raman active". This is related to the symmetry of the situation.

3. Basic parameters
   The situation under consideration here is a sphere with isotropic elasticity (the "nanoparticle") embedded in an infinite isotropic elastic medium (the "matrix"). Their properties are specified by parameters in the table below.

Table I.     Properties of nanoparticle and matrix
propertyunitsnanoparticlematrix 
density kg/m3 ρpρm
longitudinal
sound velocity 		m/s Clp Clm Clp=sqrt((λp+2μp)/ρp)
transverse
sound velocity 		m/s Ctp Ctm Ctp=sqrt(μp/ρp)
1st Lamé constant Paλp λm
Shear modulus Pa μp μm
radius mRpRmRm → ∞
diameter mDp Dp = 2Rp

   Let this composite object take the form of a sphere of radius Rm. For simplicity, we will want to eventually take the limit Rm → ∞ and in any case assume that Rp << Rm. For a coordinate system, choose the origin to be at the center of the sphere, and use spherical coordinates (r,θ,φ), related to Cartesian coordinates by x = r sin(θ) cos(φ), y = r sin(θ) sin(φ), z = r cos(θ).
   Mechanical vibrations in this solid are described in terms of a displacement field u(r,θ,φ,t) (more conveniently denoted as "u(r,t)"). The units of u are metres. u is the displacement r' - r of a point of material whose equilibrium position is r. For greatest simplicity, the boundary conditions are assumed to be that
u(Rm,θ,φ,t) = 0 for all θ and φ
that is, the surface of the sphere is rigidly fixed.

Article on Si photoelasticity pointed out to me by S. P. Wong
H. Liang, Y. Pan, S. Zhao, G. Qin and K. K. Chin, "Two-dimensional state of stress in silicon wafer", J. Appl. Phys. 71 (1992) pages 2863-2870
Ken K. Chin web page: http://physics.njit.edu/phys/Fac-Staff/People/Chin.html
Ken K. Chin Email: [email protected]

H. J. Peng, S. P. Wong and S. Zhao, "Photoelastic study of stress field under thin oxide film edge in silicon and the validity of the concentrated force model" J. Phys. D: Appl. Phys. 35 (2002) L1-L3 ::
S. P. Wong, H. J. Peng, S. Zhao, "Analytic solution of stress distribution under a thin film edge in substrates" Applied Physics Letters (Appl. Phys. Lett.) vol. 79 (2001) page 1628 ::
S. P. Wong, W. Y. Cheung, N. Ke, M. R. Sajan, W. S. Guo, L. Huang and S. Zhao, "IR photoelasticity study of stress distribution in silicon under thin film structures" Materials Chemistry and Physics 51 (1997) 157-162 ::


Sorry, but
This page is still under construction.
   Apart from quantum mechanical corrections, a linear mechanical system in thermal equilibrium at temperature T (absolute temperature, Kelvin degrees) will have average energy kBT for each mechanical degree of freedom. kB is Boltzmann's constant, 1.381×10-23 J/K. This is commonly called the "equipartition theorem". Apply this to a linear elastic continuum solid (in particular, the glass "matrix" surrounding a nanoparticle) with density ρm, shear modulus μm, longitudinal speed of sound clm and transverse speed of sound ctm. Realistic properties of the material, such as anisotropy of its elasticity or dispersion of phonon modes at higher frequencies, are ignored in everything that follows.    Since the system is linear, such a general disturbance is the linear combination of normal modes of vibration of the form
u(r,t) = cos(ωt) u(r)
where ω is the angular frequency of the vibration (in radians per second).
   The next job is to come up with a orthogonal basis of functions corresponding to vibrational modes with angular frequency ω. Vibrational normal modes in an isotropic continuous solid can be longitudinal or transverse. Transverse vibrations have two possible directions of polarization.

Case I: Longitudinal ("Spheroidal") Modes
    The longitudinal modes (often called "spheroidal" modes) have the form (for integer l = 0, 1, 2, 3...)
u(r,t) = cos(ωt) ∇( A jl (klm r) Pl (cos(θ)) )
ω = angular frequency (radians/second)
∇ = gradient differential operator
A = amplitude (real number, in metres squared)
klm = ω / clm    ("l " = "longitudinal", "m" = "matrix")
clm = longitudinal speed of sound in matrix material
jl (.) = spherical Bessel function of first kind of order l       (See: mathworld.wolfram.com)
Pl (.) = Legendre polynomial of order l

Evaluating the gradient,
ur = cos(ωt) A (d/dr) jl (klm r) Pl (cos(θ))
uθ = cos(ωt) A (1/r) (d/dθ) jl (klm r) Pl (cos(θ))
In the large-r limit, only ur makes a significant contribution to the energy, so uθ will be ignored.
The velocity is
vr = - ω sin(ωt) A (d/dr) jl (klm r) Pl (cos(θ))
   A useful identity for spherical Bessel functions of the first kind is (from: mathworld.wolfram.com)
jn(x) = (-1)n (x)n ( (1/x) d/dx )n (sin(x)/x)
while for spherical Bessel functions of the second kind
nn(x) = (-1)n (x)n ( (1/x) d/dx )n (-cos(x)/x)
(In particular, j0(x) = sin(x)/x, j1(x) = sin(x)/x2 - cos(x)/x, n0(x) = -cos(x)/x, n1(x) = - cos(x)/x2 - sin(x)/x. )
   If x is large, so that the leading term goes as 1/x, then approximately
jn(large x) =   sin(x) / x n = 0, 4, 8, ...
jn(large x) = -cos(x) / x n = 1, 5, 9, ...
jn(large x) = -sin(x) / x n = 2, 6, 10, ...
jn(large x) =   cos(x) / x n = 3, 7, 11, ...
   In what follows, the sign of the velocity is not important. That is because we are only interested in the energy. Also, whether it is cos(x) or sin(x) is not important. That is because of the large r limit being taken. In this spirit, the approximate form for the velocity field r component is
vr = - ω sin(ωt) A (d/dr) ( cos(klm r)/(klm r) ) Pl (cos(θ))
Taking the r derivative, and keeping the dominant term only
vr = ω sin(ωt) A klm sin(klm r)/(klm r) Pl (cos(θ))
vr = ω sin(ωt) A sin(klm r)/(r) Pl (cos(θ))
   Kinetic energy density (joules per cubic metre) is uKE = ½ ρm v2. The time-averaged, local-space-averaged kinetic energy density is
(0.5) ρm ω2 0.5 A2 0.5 r -2 (Pl (cos(θ)))2
The time-averaged, local-space-averaged total energy density (including kinetic and potential energy) is
(0.5) ρm ω2 0.5 A2 r -2 (Pl (cos(θ)))2
The total energy of the vibrational mode is obtained by integrating this over the entire volume (0≤r≤Rm), which involves introducing the differential volume factor, dV = r 2 sin(θ) dθ dφ dr.
Let Fl represent the integral over θ, from 0 to π, of
Fl = ∫ sin(θ) ( Pl (cos(θ)) )2 dθ
Based on standard properties of Legendre polynomials (Introduction to Electrodynamics 3rd Edition, D. Griffiths, page 140), Fl = 2/(2l+1). I also verified this numerically with a short C++ program finfl.cpp.
The φ part of the integral simply introduces a factor of 2π.
The energy of the vibrational mode then is the integral from 0 to Rm of
U = ∫ (0.5) ρm (0.5) ω2 2π A2 Fl r 2 r -2 dr
which equals the integral from 0 to Rm of
U = ∫ (0.5) ρm π ω2 A2 Fl dr
which is:
U = (0.5) ρm π ω2 A2 Fl Rm
In thermal equilibrium, (ignoring quantum effects) this equals kBT, so therefore,
kBT = (0.5) ρm π ω2 A2 Fl Rm
and consequently,
A2 = kBT / ( (0.5) ρm π ω2 Fl Rm )
In particular, A varies as one over the frequency of the mode.
   The frequency of individual modes is determined by the boundary condition, u = 0 at r = Rm. This means that
sin(klm Rm) = 0      (if l is even)
cos(klm Rm) = 0      (if l is odd)
   In any case, the allowed frequencies are approximately of the form
ω Rm / clm = n π
where n is a positive integer.
ω = n π clm / Rm
   Consider a range of frequencies of width dω. The number of modes with frequencies in this range is
n = dω Rm / (π clm )
Suppose I want to represent all normal modes in the frequency range dω by a single function with amplitude A. I want this function to have a total energy n kBT. (Note that this function represents a combination of normal modes). The amplitude of this function is obtained from:
A2 = (dω Rm / (π clm )) kBT / ( (0.5) ρm π ω2 Fl Rm )
   It is at this point that Rm cancels out, and we are easily able to take the macroscopic (i.e. thermodynamic) limit that Rm is large. In that case,
A2 = dω kBT / ( (0.5) π2 ρm clm Fl ω2)
A2 = dω kBT (2 l + 1) / (π2 ρm clm ω2)
Note that there is frequency dependence of this amplitude. Note that the spherical symmetry of the problem means that there are actually 2l + 1 vibrational modes of this type, corresponding to different values of the azimuthal quantum number m = -l, .., l.


Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca

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